5 UPPER AND LOWER INTEGRALS 117
For each integer N g> 1 it follows from (13.5.7) that
N N
\n=l / n=l
N
ow apply (12,5.7) to the increasing sequence of partial sums
n= 1
If (/n) is any sequence of mappings of X into R, the mapping
x h^lim inf /„(*) (resp. x h^lim sup/n(x))
n-*oo n-*oo
j defined for all x e X. It is denoted by lim inf/w (resp. lim sup/n).
H-+QO «-*oo
13.5,9) (Fatou's lemma) If(fn) is any sequence of functions ^0, then
M*(lim inf/.) glim inf /im
\ i»-»oo / n-+oo
For each n ^ 1, let gn = inf (fn+p). Clearly we have n*(gn] ^ inf ^*(/n+p
1>£0 p^O
nd ^n ^0; since the sequence (0rB) is increasing and liminfyj, = sup^nj it
ollows from (13.5.7) that
^*/lim inf/n\ = sup v*(gn} ^ sup (inf jU*(
\ n-»oo Jn n \p^0
If/: X ~> K is any mapping, we define /**(/) = — /**(-/); this number is
ailed the /0vtw integral of /with respect to the measure /i. All the properties
.bove which were proved for the upper integral can be immediately translated
tito properties of the lower integral. In particular, if we put — J = ^ (or
^(X)), then y is the set of all upper semicontinuous functions on X which are
Bounded above by a function belonging to «5TR(X) (which implies that they do
lot take the value + oo). For all functions /e £f, we have
ind for all/: X -* R we have
= sup
[1 3.5.10) Iff is any mapping ofX into I, then /**(/) <; /i*(/). otherwise there is